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CS 140 Lecture 4

CS 140 Lecture 4. Professor CK Cheng 4/11/02. Part I. Combinational Logic. Implementation K-Map Given F R D Obj: Minimize sum of products Proc: Draw K-Map Derive prime implicants Derive the essential prime implicants Derive minimum expression.

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CS 140 Lecture 4

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  1. CS 140 Lecture 4 Professor CK Cheng 4/11/02

  2. Part I. Combinational Logic Implementation K-Map Given F R D Obj: Minimize sum of products Proc: Draw K-Map Derive prime implicants Derive the essential prime implicants Derive minimum expression

  3. Example Given F = Sm (0, 3, 4, 14, 15) D = Sm (1, 11, 13) K-map b 0 4 12 8 1 1 0 0 1 5 13 9 - 0 - 0 d 3 7 15 11 1 0 1 - c 2 6 14 10 1 0 1 0 a

  4. Prime Implicants: Largest rectangles that intersect On Set but not Off Set that correspond to product terms. E.g. Sm (0, 4), Sm (0, 1), Sm (1, 3), Sm (3, 11), Sm (14, 15), Sm (11, 15), Sm (13, 15) Essential Primes: Prime implicants covering elements in F that are not covered by any other primes. E.g. Sm (0, 4), Sm (14, 15) Min exp: Sm (0, 4), Sm (14, 15), ( Sm (3, 11) or Sm (1,3) ) f(a,b,c,d) = a’b’c’ + abc’ + b’cd (or a’b’d)

  5. Corresponding circuit a’ c’ d’ f(a,b,c,d) a b c b’ c d

  6. Another example Given F = Sm (3, 5), D = Sm (0, 4) b 0 2 6 4 - 0 0 - 1 3 7 5 c 0 1 0 1 a Primes: Sm (3), Sm (4, 5) Essential Primes: Sm (3), Sm (4, 5) Min exp: f(a,b,c) = a’bc + ab’

  7. 5 variable K-map c c 0 4 12 8 16 20 28 24 1 5 13 9 17 21 29 25 e e 3 7 15 11 19 23 31 27 d d 2 6 14 10 18 22 30 26 b b a Neighbors of 5 are: 1, 4, 13, 7, and 21 Neighbors of 10 are: 2, 8, 10 ,14, and 26

  8. 6 variable K-map d d 0 4 12 8 16 20 28 24 1 5 13 9 17 21 29 25 f f 3 7 15 11 19 23 31 27 e e 2 6 14 10 18 22 30 26 c c d d 32 36 44 40 48 52 60 56 33 27 45 41 49 53 61 57 b f f 35 39 47 43 51 53 63 59 e e 34 38 46 42 50 54 62 58 c c a

  9. Min product of sums Given F = Sm (3, 5), D = Sm (0, 4) b 0 2 6 4 - 0 0 - 1 3 7 5 c 0 1 0 1 a Prime Implicates: PM (0,1), PM (0,2,4,6), PM (6,7) Essential Primes Implicates: PM (0,1), PM (0,2,4,6), PM (6,7) Min exp: f(a,b,c) = (a+b)(c )(a’+b’)

  10. Corresponding Circuit a b f(a,b,c,d) a’ b’ c

  11. Another min product of sums example Given F = Sm (0, 3, 4, 14, 15) D = Sm (1, 11, 13) K-map b 0 4 12 8 1 1 0 0 1 5 13 9 - 0 - 0 d 3 7 15 11 1 0 1 - c 2 6 14 10 0 0 1 0 a

  12. Prime Implicates: PM (2,6), PM (2,10), PM (1,5,9,13), PM (5,7), PM (6,7), PM (8,9,10,11), PM (8,9,12,13) Essential Primes: PM (8,9,12,13) Min exp: PM (8,9,12,13) PM (5,7), PM (2,6), PM (8,9,10,11) or PM (6,7), PM (1,5,9,13), PM (2,10) f(a,b,c,d) = (a+b’+d’)(a’+c’+d)(a’+b)(a’+c)

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